By Hidefumi KUBOTA
Summary
There is a problem of the
fluctuation of energy of Cooper pairs which cannot be explained by the
superconducting mechanism. But I think that it is explained by shifting of
energy between the kinetic energy of barycentric motion of the Cooper pairs and
the kinetic energy of antiparallel motion of the electrons.
At first I check the momentum and the kinetic energy of a Cooper pair (1). The state of motion of a Cooper pair and its two electrons is like the next figure.
Assume m is the mass of an electron, v is the speed of antiparallel motion of the electron, V and V ' are the speed of the electrons in the direction which a permanent current flows through.
Then the momentum of the Cooper is
mv－mv+mV+mV '=m(V+V ').
The kinetic energy of barycentric motion of the Cooper pair is
mV2/2+mV '2/2=m(V2+V '2)/2.
The kinetic energy of antiparallel motion of the Cooper pair is
mv2/2+mv2/2=mv2.
Accordingly all the kinetic energy of the Cooper pair is
mv2+m(V2+V '2)/2.
Next, the superconductive state is generated by Cooper pairs condensing into a same momentum (2). Namely, in the superconductive state, the size of momentum “m(V+V ')” of Cooper pairs in a system must be same. That means that energy to compensate the decrease of momentum is supplied by a mechanism characteristic of superconductivity when a Cooper pair loses energy by lattice vibration and collision with lattices and impurities and its momentum decreases.
Assume a certain electron acts on neighboring lattices and lost energy by the influence of Coulomb force. Since the lattices are attracted towards the electron, the local area has higher density of positive electric charge than the other areas. Then a different electron is attracted by the local area of the higher density of positive electric charge and is accelerated and can get energy. The superconductive state uses such relations. In the superconductive state, electrons make Cooper pairs, and the one electron gets energy to compensate the energy that the other electron of the pair lost, and the state of a same momentum is kept (3). This mechanism is the first superconducting mechanism. In this way the state of same size of momentum of Cooper pairs is kept even though lattice vibration acts on the Cooper pairs.
This mechanism is premised on the lattice vibration generated by Coulomb force. Since the cases of lattices and impurities are physical collision, they lack in this premise. Lattice vibration is surely main cause of electrical resistance. However, it has no mistake that collisions with lattices and impurities cause electrical resistance (3). Therefore, in this situation the kinetic energy of Cooper pairs to collide with lattices and impurities will be lost, they will be piled up, and the permanent current will be attenuated. In order to prevent it, a mechanism similar to the first superconducting mechanism to compensate for the energy has to work on the collision with lattices and impurities.
The mechanism that I discovered is following. The barycentric motion of Cooper pairs is the substance of a permanent current. I am taking up the problem that kinetic energy of this barycentric motion is lost by collisions with lattices and impurities. Other than this barycentric motion, each electron of a Cooper pair is doing antiparallel motion, and the Cooper pair has kinetic energy “mv2” of this antiparallel motion. I think that the kinetic energy of this antiparallel motion changes into kinetic energy “m(V2+V '2)/2” of the barycentric motion. By this change the barycentric motion gets energy to compensate for the lost energy by collisions and the state of a same momentum is kept.
There are causes to disturb the state of energy of Cooper pairs other than lattice vibration and lattices and impurities. They are the cases that only a part of Cooper pairs in a system are given energy of an electric field and/or a magnetic field. Of these cases, I will try to consider the case that only a part of Cooper pairs in a system were given energy of a magnetic field.
The direction of electromagnetic force generated on electrons by a magnetic field is perpendicular to the direction of moving electrons. Therefore, there is no change of momentum and kinetic energy of the barycentric motion of the Cooper pairs in the direction that the permanent current flows through even though a magnetic field works on the Cooper pair. And by this the magnetic field never changes the sizes of permanent currents. However, as for the whole Cooper pair, the momentum “m(V+V ')” of barycentric motion of the Cooper pair would increase.
Since the superconductive state is generated by Cooper pairs condensing into a same momentum, this increase must change into another form. I think that this change takes a form to change kinetic energy of the barycentric motion into kinetic energy of antiparallel motion.
As above, there is a problem of the fluctuation of energy of Cooper pairs which cannot be explained by the first superconducting mechanism. I think the problem can be explained by shifting of energy between kinetic energy “m(V2+V '2)/2” of barycentric motion of Cooper pairs and kinetic energy “mv2” of antiparallel motion.
The kinetic energy of antiparallel motion is much bigger than kinetic energy of barycentric motion, but is not infinite. Since energy to compensate for the loss of kinetic energy of barycentric motion is not infinite, permanent currents are not eternal. However, it may be said that they are nearly eternal because kinetic energy of antiparallel motion can increase by getting energy from such as a magnetic field.
References
(1) Taiichiro OHTSUKA, Chyodendo no Sekai [World of Superconductivity]. Tokyo: Kodansha, 1987
(2) A. C. Rose-Innes, E. H. Rhoderick, Chyodendo Nyuumon [Introduction to Superconductivity Second Edition]. Tokyo: Sangyo Tosho, 1978
(3) Masato MURAKAMI, Yasashii Chyodendo no Hanashi [Plain Explanation on Superconductivity], http://moniko.s26.xrea.com/cyoudendou_kiso.htm